The current that squeezes what carries it

The field that wraps a current found that Ampère’s law counts current: go once round a closed path, add up the magnetic field along it, and the answer is $\mu_0$ times the current threading the path. The later arguments followed the law into solenoids, into matter and into a scalar potential, and the force between pieces of a current found that the force one element of current exerts on another cannot be defined piece by piece, only for whole circuits. In every case the current ran in wires that did not move.

Let the conductor move and the law acquires a mechanical consequence. A current flowing along a column makes a field that circles the column, and every element of the current, crossing that field, is pushed towards the axis. Parallel currents attract. In a copper wire the push is resisted by the metal and goes unnoticed. In a column of plasma there is nothing to resist it but the plasma’s own pressure, and the current squeezes the column until the two balance. That balance turns out to be one of the most economical results in plasma physics: it fixes the current a hot column needs from two numbers, and leaves out everything else one might expect to matter.

A tube that lightning crushed #

The effect was seen before anyone was looking for it. In 1905 two physicists at the University of Sydney, Pollock and Barraclough, were sent a length of copper tube from a lightning conductor at Hartley Vale in New South Wales. After a strike it had been found squashed: its round cross-section flattened, as though something had crushed it from outside. Nothing had touched it. They worked out that the lightning current, flowing along the tube, had made a field round it, and that the field had pushed the tube’s own walls inward hard enough to buckle them.

The push is the magnetic pressure of the same force whichever way the surface faces, $B^2/2\mu_0$, applied at the tube’s outer surface where the field is $\mu_0 I/2\pi r$. It grows as the square of the current and falls as the square of the radius.

A typical lightning stroke carries thirty thousand amperes. On a tube a centimetre in radius that is a squeeze of 1.4 atmospheres, enough to dent thin sheet if it lasted, but it lasts tens of microseconds. The strongest strokes reach two hundred thousand amperes, and on the same tube that is sixty-four atmospheres; on a tube half the radius it is two hundred and fifty. A thin-walled copper tube cannot carry that inward load for a hundred microseconds without buckling, and the tube from Hartley Vale had been hit by such a stroke. It is generally taken as the first recorded observation of what was later called the pinch effect.

Balancing a column against its own field #

A plasma column carrying a current along its axis is the same geometry with the walls replaced by gas. At every radius the plasma’s pressure pushes outward and the current density, crossing the azimuthal field, pushes inward. Equilibrium needs

$$\frac{dp}{dr} = -J_z B_\theta,$$

the radial gradient of pressure balanced by the force density $J \times B$. Ampère’s law ties the field to the current enclosed: $B_\theta(r) = \mu_0 I(r)/2\pi r$, with $I(r)$ the current flowing inside radius $r$.

The figure draws one such equilibrium, the profile Bennett himself found in 1934 for a column in which electrons and ions drift with fixed speeds. The pressure is highest on the axis, 1.3 gigapascals for a column of hydrogen at ten kiloelectronvolts holding $10^{19}$ ions in each metre — thirteen thousand atmospheres — and falls outward. The enclosed current rises from zero at the axis to its total of 0.8 megaamperes. The field rises linearly near the axis, where the enclosed current grows as the area, and falls as $1/r$ outside the dense core, where there is little current left to add. It peaks at twenty-nine tesla. At every radius the pressure gradient and the magnetic force cancel.

The integral that forgets the shape #

The surprising thing is what happens when the force balance is integrated across the whole column. Multiply both sides by $r^2$ and integrate from the axis to the edge, where the pressure has fallen to zero. On the magnetic side, Ampère’s law lets $J_z B_\theta r^2$ be written as the derivative of $(\mu_0 I(r)^2)/(8\pi^2)$, so it integrates to its value at the edge, which depends only on the total current. On the pressure side, an integration by parts turns $\int r^2\, dp$ into $-2\int r\,p\,dr$, which is the total pressure per unit length. The result is Bennett’s relation:

$$\frac{\mu_0 I^2}{8\pi} = \int_0^R 2\pi r\, p\, dr = N k (T_e + T_i),$$

where $N$ is the number of ions in each metre of column, equal to the number of electrons, and $T_e$ and $T_i$ are their temperatures. The radius of the column does not appear. Neither does the profile of the pressure, nor where the current flows inside the column. Only the total current, the number of particles per metre and the temperature remain.

The figure tests the claim the hard way. It takes four columns that look nothing alike: one with the plasma spread evenly to the edge, one parabolic, one sharply peaked on the axis and one hollow, with the plasma concentrated in a ring and the axis nearly empty. Each holds the same particles at the same temperature, which means the same area under its pressure profile weighted by the radius. For each one the field is found from the force balance, radius by radius, and the total current is read off at the edge. All four need 0.800 megaamperes. The peaked column needs a field of twenty-nine tesla inside it and the flat one a field that grows steadily to the edge, but the total current, which is what a power supply delivers and what can be measured with a coil outside the column, is the same.

The reason is a version of the virial theorem. Multiplying by $r^2$ and integrating is exactly the operation that turns a force balance into a balance of energies: the plasma’s thermal energy per unit length against the magnetic energy the current stores, and for an equilibrium those two are fixed in ratio whatever the distribution. It is the same reason the pressure inside a star can be estimated from its mass and radius alone without knowing how its density is distributed.

What a column needs at every temperature #

Bennett’s relation gives the confining current directly. For hydrogen with electrons and ions at the same temperature, $I = \sqrt{16\pi N k T/\mu_0}$.

The lines are parallel because the current goes as the square root of the product of the line density and the temperature. A column hot enough for fusion, ten kiloelectronvolts, holding $10^{19}$ ions per metre needs eight hundred thousand amperes. A spark channel in air, much denser but at only one or two electronvolts, needs about a hundred thousand, which is why the strongest lightning strokes can begin to pinch their own channels while ordinary ones are held by the surrounding air. A glow discharge in a fluorescent tube, at ten electronvolts and $10^{17}$ ions per metre, would need three thousand amperes. It carries a fraction of an ampere, so its magnetic squeeze is negligible and its plasma is held by the glass.

The relation also says what changing the current does. Raise the current on a column in equilibrium and the right-hand side must rise to match: either the column heats, or it gathers more particles, or, if it can do neither fast enough, it is out of balance and contracts. The contraction compresses the plasma, which heats it, and a column driven by a rising current is both squeezed and heated by the same current — the scheme by which the early fusion experiments of the 1950s hoped to reach fusion temperatures without any other heating.

Squeezing a column by turning up the current #

The relation also says how hard a rising current compresses. Suppose the current climbs slowly enough for the column to stay in balance, and fast enough that no heat escapes and no particles are added. Then the plasma is compressed adiabatically, and for a column squeezed sideways at fixed length its temperature rises as its area falls to the power $\gamma - 1$, which for a hydrogen plasma is two-thirds: halve the radius and the temperature goes up by $2^{4/3}$, about 2.5. Bennett’s relation, with $N$ fixed, says the temperature must also go as $I^2$. Putting the two together, the radius falls as $I^{-3/2}$.

Doubling the current therefore shrinks the column by a factor of 2.8, raises its density eightfold and its temperature fourfold. The fusion power a column produces per unit length goes as the square of the density times the area times a reaction rate that climbs steeply with temperature, so each doubling of the current multiplies it many times over. That is the arithmetic that made the pinch the first candidate for controlled fusion in the early 1950s, before anything was known about its stability: it promised heating and compression together, from one power supply, with no external magnet at all.

It also explains why a pinch is the most economical magnetic container there is. The ratio of plasma pressure to magnetic pressure, called beta, measures how much of the field a configuration actually spends on holding plasma. A tokamak runs at a few per cent, because almost all of its field is the strong axial field that keeps the column stable and does no confining. In a pinch the only field is the one the current makes, the plasma pressure at the axis equals the magnetic pressure at the edge, and beta is of order one. Every tesla the current produces is pushing on plasma. The price, as the instabilities below show, is that nothing is left over to keep the column straight.

Why the radius is left free #

That the radius is absent from the balance is the most consequential part of it. It means a column in Bennett equilibrium is equally balanced at any radius: squeeze it to half its width at the same temperature and the pressure rises fourfold, the field rises twofold, the magnetic pressure fourfold, and nothing has changed in the balance. The equilibrium does not choose a radius, and so it does not resist being pushed to a different one.

A column that does not resist a change of radius uniformly does not resist one locally either. Squeeze it at one place along its length and the field there, which goes as one over the radius, rises, which squeezes it further: the column necks into a string of beads, the sausage instability. Bend it slightly and the field lines crowd on the inside of the bend, where the magnetic pressure is higher, which pushes the bend further out: the kink instability. A bare pinch is unstable to both, within microseconds, and the history of fusion research in the 1950s is largely the history of finding this out. The price is paid here. The cure that worked was to add a strong magnetic field along the column, which resists both bending and necking, and to bend the column into a ring so that it has no ends. That arrangement, with the axial field much larger than the field of the current, is the tokamak, and the condition it imposes on the twist of the field lines — the safety factor — is the stability condition for the kink. What happens when a pinch is driven so hard that it relaxes through its instabilities is the subject of the twist that outlives the turbulence.

The current at which a pinch radiates itself away #

Bennett’s relation has one more consequence, and it is a ceiling rather than a floor. A hot column of hydrogen is heated by its own current, through the plasma’s resistance, and it loses energy by bremsstrahlung, the radiation electrons give off when they are deflected by ions, as a charge that turns must glow requires. Both rates can be computed, and Bennett’s relation then removes every variable but one.

The ohmic heating per unit length is the resistivity times $I^2/\pi a^2$, and a plasma’s resistivity falls as $T^{-3/2}$: a hotter plasma has faster electrons, which collide less. The bremsstrahlung per unit length goes as the square of the density times $\sqrt{T}$ times the area, which is $N^2\sqrt{T}/\pi a^2$. Their ratio is therefore proportional to $N^2 T^2/I^2$, with the radius cancelled. Bennett’s relation says $NT$ is proportional to $I^2$, so the ratio is proportional to $I^2$ alone.

The ratio reaches one at a current independent of the column’s density, temperature and size: about 1.1 to 1.6 megaamperes for hydrogen, depending on the slowly varying logarithm that enters the resistivity. This is the Pease–Braginskii current, found independently by Pease in Britain and Braginskii in the Soviet Union in 1957. Below it, a column in equilibrium is heated faster than it radiates and warms. Above it, it radiates faster than it is heated, cools, and — since Bennett’s relation then requires less pressure than the current is exerting — is squeezed. The squeeze raises the density, the radiation rises with the square of the density, and the column collapses towards a thin, dense, radiating thread, until something the model leaves out — the opacity of the plasma to its own radiation, the quantum pressure of its electrons, an instability — stops it.

Pinches driven far above that current are among the brightest laboratory sources of X-rays, and are built to be. The Z machine at Sandia drives currents of over twenty megaamperes through arrays of fine wires, which vaporise into a plasma shell that the current implodes onto the axis, radiating megajoules of X-rays in a few nanoseconds. Radiative collapse, a failure for a fusion pinch, is a design goal there.

A current that is also a container #

The same arithmetic applies to anything carrying a current that is free to move. A beam of electrons travelling through a gas, once it has driven out enough of the gas’s electrons to cancel its own charge, is held together by its own magnetic field, which is the case Bennett was actually studying in 1934 and the one the beam that stops pushing itself apart followed from the electric side. A beam of protons, a jet of plasma from a star, a current sheet in a planetary magnetosphere all pinch in proportion to the square of their current. The relation even survives in a solid: the electrons and holes injected into a bar of indium antimonide, carrying a large current, were seen in the early 1960s to crowd towards its axis as a pinched electron–hole plasma.

What is striking in every case is how little the balance needs to know. The current no particle carries found that the current holding a magnetised plasma against its own pressure is a statement about the pressure gradient, not about any particle’s motion. Bennett’s relation is the integral of that statement across a whole column, and in the integral the gradient’s shape disappears. What is left is an energy balance in which the magnetic energy of the current matches the thermal energy of the plasma it carries.

What the pictures cannot show #

The figures show equilibria, and the equilibria they show are unstable: a real pinch necks and kinks within a few microseconds unless something stabilises it, and the profiles drawn here describe a column for at most that long. They assume a straight, infinitely long column with no axial field and no end effects. They take the electrons and ions at one temperature, which in a fast pinch they are often not. The radiation figure uses the bremsstrahlung of a hydrogen plasma transparent to its own radiation; impurities with more charge radiate far more strongly and lower the collapse current, and an opaque plasma traps its radiation and raises it. And Spitzer’s resistivity assumes a plasma close to equilibrium, which a collapsing pinch is not.

Still open: whether a pinch can be made stable enough to burn #

The simplest fusion scheme of all is a straight Z-pinch with no axial field, and it was abandoned in the 1950s because it was unstable. It has been reopened. Experiments since the late 2010s have stabilised the column against the kink and sausage modes by making the plasma flow along the axis with a velocity that varies with radius — the shear prevents the instabilities from growing coherently — and have held columns for tens of microseconds, many times the growth time of the instabilities, at kiloelectronvolt temperatures. Whether sheared flow can keep a column stable at the currents and densities Bennett’s relation demands for fusion conditions, and whether it can be scaled to a power plant, is being tested now.

The habit worth carrying away is to integrate a balance before solving it. A force balance multiplied by the right power of the radius becomes an energy balance, and the energy balance of a current-carrying column fixes the current from the particles per metre and their temperature alone — $\mu_0 I^2/8\pi = Nk(T_e + T_i)$ — independent of the column’s radius and of how its plasma is arranged. Four columns that look nothing alike need 0.80 MA each, and a hydrogen column driven past about 1.4 MA radiates faster than it is heated and collapses.